A characterisation of the planes meeting a non-singular quadric of PG(4,Q) in a conic

Abstract

By counting and geometric arguments, we provide a combinatorial characterisation of the planes meeting the non-singular quadric of PG(4,q) in a conic. A characterisation of the tangents and generators of this quadric when q is odd has been proved by de Resmini [15], and we give an alternative using our result.

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References

  1. [1]

    A. Barlotti: Un’estenstione del teorema di Segre-Kustaanhemo, Boll. Un. Mat. Ital. 10 (1955), 96–98.

    MathSciNet  MATH  Google Scholar 

  2. [2]

    S. G. Barwick and D. K. Butler: A characterisation of the lines external to an oval cone in PG(3,q), q even, to appear in J. Geom.

  3. [3]

    A. Bichara and C. Zanella: Tangential Tallini sets in finite Grassmannians of lines, J. Combin. Theory Ser. A 109 (2005), 189–202.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    D. K. Butler and S. G. Barwick: A characterisation of the lines external to a quadric cone in PG(3,q), q odd, preprint

  5. [5]

    R. C. Bose and R. C. Burton: A characterization of at spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes, J. Combinatorial Theory 1 (1966), 96–104.

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    F. Buekenhout: Ensembles quadratiques des espaces projectifs, Math Z. 110 (1969), 306–318.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    R. Di Gennaro, N. Durante and D. Olanda: A characterization of the family of lines external to a hyperbolic quadric of PG(3,q), J. Geom. 80 (2004), 65–74.

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    N. Durante and D. Olanda: A characterization of the family of secant or external lines of an ovoid of PG(3,q), Bull. Belg. Math. Soc. Simon Stevin 12 (2005), 1–4.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    J. W. P. Hirschfeld: Projective geometries over finite fields, 2nd Edition, Oxford University Press, Oxford, 1998.

    Google Scholar 

  10. [10]

    J. W. P. Hirschfeld: Finite Projective Spaces of Three Dimensions, Oxford University Press, Oxford, 1985.

    Google Scholar 

  11. [11]

    J. W. P. Hirschfeld and J. A. Thas: General Galois Geometries, Oxford Univ. Press, Oxford, 1991.

    Google Scholar 

  12. [12]

    C. M. O’Keefe: Ovoids in PG(3,q): a survey, Discrete Math. 151 (1996), 171–188.

    Google Scholar 

  13. [13]

    M. Oxenham and Rey Casse: Towards the Determination of the Regular n-covers of PG(3,q), Boll. Unione Mat. Ital. Sez B Artic. Ric. Mat. 6(8) (2003), 57–87.

    MathSciNet  MATH  Google Scholar 

  14. [14]

    G. Panella: Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito, Boll. Un. Mat. Ital. 10 (1955) 507–513

    MathSciNet  MATH  Google Scholar 

  15. [15]

    M. J. de Resmini: A characterization of the set of lines either tangent to or lying on a nonsingular quadric in PG(4,q), q odd, Finite Geometries (Winnipeg, Man., 1984), 271–288, Lecture Notes in Pure and Appl. Math. 103 Dekker, New York (1985).

  16. [16]

    J. A. Thas: A combinatorial problem, Geom. Dedicata 1 (1973), 236–240.

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to David K. Butler.

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Butler, D.K. A characterisation of the planes meeting a non-singular quadric of PG(4,Q) in a conic. Combinatorica 33, 161–179 (2013). https://doi.org/10.1007/s00493-013-2402-7

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Mathematics Subject Classification (2000)

  • 51E20